\subsection{可达速率域内界}\label{Cha:BC:SISO:InnerBound}
根据信息论，对于第$ i $个用户，可达速率$ R^{\mathrm{BC,SISO}}_i $的下界为
\begin{subequations}
    \begin{align}
    R^{\mathrm{BC,SISO}}_i &=\maximize{\left\{f_{X_i}\left(x_i\right) \right\}}{I\left(X_i;Y_i\right)}\\
    &=\maximize{\left\{f_{X_i}\left(x_i\right) \right\}}{h\left(Y_i\right)-h\left(Y_i\vert X_i\right)}\\
    &=\maximize{\left\{f_{X_i}\left(x_i\right) \right\}}{h\left(g_i\sum_{j=1}^{K}X_j+g_i b+Z_i\right)-h\left(g_i\sum_{j=1,j\neq i}^{K}X_j+g_i b+Z_i\right)}\\
    &\geq \maximize{\left\{f_{X_i}\left(x_i\right) \right\}}{\frac{1}{2}\log_2\left(\sum_{j=1}^{K}2^{2 h\left(g_i X_j\right)}+2^{2 h\left(Z_i\right)}\right)}\nonumber\\
    &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{1}{2}\log_2 2\pi e \var{g_i\sum_{j=1,j\neq i}^{K}X_j+g_i b+Z_i}\label{Eqn:BC:SISO:InnerBound:d}\\
    &=\maximize{\left\{f_{X_i}\left(x_i\right) \right\}}{\frac{1}{2}\log_2\left(g_i^2\sum_{j=1}^{K}{2^{2 h\left( X_j\right)}}+2\pi e \sigma^2\right)}\nonumber\\
    &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{1}{2}\log_2{\left(2\pi e g_i^2\sum_{j=1,j\neq i}^{K}\nu_j\varepsilon+2\pi e \sigma^2\right)},\label{Eqn:BC:SISO:InnerBound:e}
    \end{align}
\end{subequations}
式中，不等式\eqref{Eqn:BC:SISO:InnerBound:d}是根据熵功率不等式，以及对于给定方差$ \var{Q} $的任意随机变量$ Q $，有$ h\left(Q\right)\leq \frac{1}{2}\log_2{2\pi e \var{Q}} $。

由式\eqref{Eqn:BC:SISO:InnerBound:e}的单调性知，通过最大化微分熵$ \left\{h\left(X_j\right)\right\} $，可以最大化下界\eqref{Eqn:BC:SISO:InnerBound:e}。将定理\ref{Thm:P2P:SISO:Lower:ABG}的ABG分布的微分熵\eqref{Eqn:P2P:SISO:Lower:MaxEntropy:EntropyVal}代入至\eqref{Eqn:BC:SISO:InnerBound:e}中，得
    \begin{align}
    R^{\mathrm{BC,SISO}}_i \geq = \frac{1}{2}\log_2{\left(\frac{2\pi\sigma^2+g_i^2\sum_{j=1}^{K}e^{1+2\left(\alpha_j+\gamma_j\nu_j\varepsilon\right)}}{2\pi\sigma^2+2\pi g_i^2\sum_{j=1,j\neq i}^{K}\nu_j\varepsilon}\right)}.\label{Eqn:BC:SISO:InnerBound:ABG}
    \end{align}



因此，可见光广播信道可达速率域的内界（ABG内界）可以表示为
\begin{align}
\begin{cases}
R^{\mathrm{BC,SISO}}_1 &\leq \frac{1}{2}\log_2{\left(\frac{2\pi\sigma^2+g_1^2\sum_{j=1}^{K}e^{1+2\left(\alpha_j+\gamma_j\nu_j\varepsilon\right)}}{2\pi\sigma^2+2\pi g_1^2\sum_{j=2}^{K}\nu_j\varepsilon}\right)},\\
&\vdots\\
R^{\mathrm{BC,SISO}}_K &\leq\frac{1}{2}\log_2{\left(\frac{2\pi\sigma^2+g_K^2\sum_{j=1}^{K}e^{1+2\left(\alpha_j+\gamma_j\nu_j\varepsilon\right)}}{2\pi\sigma^2+2\pi g_K^2\sum_{j=1}^{K-1}\nu_j\varepsilon}\right)}.
\end{cases}
\end{align}